Question

1. angle a is between 0° and 180°. determine all measures of angle a in the following cases. round 1 d.p 4 marks

a) sin a = 0.2458
b) cos a = - 0.6546
c) tan a = - 1.1945

1. find the measure of side a to the nearest tenth (1 decimal place). 3 marks

Explanation:

Step1: Find angle C

In a triangle, the sum of angles is \(180^{\circ}\). Given \(A = 39^{\circ}\) and \(B=65^{\circ}\), then \(C=180^{\circ}-(39^{\circ}+65^{\circ}) = 76^{\circ}\).

Step2: Apply the sine rule

The sine rule is \(\frac{a}{\sin A}=\frac{b}{\sin B}\). Here \(b = 30\) ft, \(A = 39^{\circ}\), \(B = 65^{\circ}\). So \(a=\frac{b\times\sin A}{\sin B}\).
Substitute the values: \(a=\frac{30\times\sin(39^{\circ})}{\sin(65^{\circ})}\).
We know that \(\sin(39^{\circ})\approx0.6293\) and \(\sin(65^{\circ})\approx0.9063\).
Then \(a=\frac{30\times0.6293}{0.9063}=\frac{18.879}{0.9063}\approx20.8\) ft.

Answer:

\(a\approx20.8\) ft